# Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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### dimension of linear system and multiplicity at a point

Prove the following statement. Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ then $|L|$ contains a curve $C$ passing ...
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### Existence of a global section of $L(D)$ vanishing on $D$.

Given a complex surface $X$ and $D$ a smooth curve on $X$, there is an associated line bundle $L(D)$ over $X$. Is there always a global section $s\in H^0(X,L(X))$, s.t. $s$ is just vanishing on $D$?
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### Metric on the dual line bundle

Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am struggling to understand how one induces a canonical dual metric $h^*$ on $L^*$. Now ...
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### Definition of index of a line bundle section is well-defined?

I'm reading lecture notes on advanced complex analysis and I'm struggling with a certain claim about the definition of the index of a complex line bundle section. Let $L\to M$ be a complex line bundle ...
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### Complex line bundles over a Riemann surface can be given a holomorphic line bundle structure

In p.20 of this lecture note (link: http://www.math.ubc.ca/~cautis/math428/notes-bundles.pdf), it is written that every complex line bundle over a Riemann surface can be given a holomorphic line ...
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### Number of sections of line bundles on a plane curve

Let $A$ be a line bundle of degree $1$ on a smooth plane quintic curve over complex numbers. We know that it can have at most $1$ section. My question is the following : Under what conditions it has ...
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### Connection 1-form, symplectic potential?

I recently encountered a formula that fell a little bit from the sky: Given: A symplectic manifold $(M,\omega)$, a Hermitian line bundle $\pi:B\rightarrow M$, a connection $\nabla$ on $B$ and a ...
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### How can I use Grothendieck--Riemann--Roch theorem?

For my research, I'm trying to understand how the Grothendieck--Riemann--Roch theorem is used in the paper The Birational Geometry of the Hilbert Scheme of Points on Surfaces by Aaron Bertram & ...
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### Projective bundle formula for sheaf cohomology

Given a projective bundle $p:P(E)\rightarrow X$ associated to a vector bundle $E$ and a line bundle on $P(E)$ of the form $\mathcal{O}_{P(E)}(1)\otimes p^*L$ where $L$ is a line bundle on $X$, is ...
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### Is it possible for the pullback of an ample line bundle under projection to be big?

Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line ...
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### Pullback of hermitian line bundle

let $f:X\to Y$ be a holomorphic map between complex manifolds. Let $(L,h)$ be a hermitian line bundle on $Y$. Namely $L$ is a holomorphic line bundle on $Y$ and $h$ is a $C^{\infty}$-hermitian inner ...
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### Application of gonality of a plane curve

Let $X$ be a smooth plane projective curve (over $\mathbb C$) of degree $5$ and genus $6$. Then we know that it has gonality $4$. This means the minimum degree of a base point free line bundle with ...
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### How to see this identification of a Springer fiber with the bundle $\mathcal{O}_{\mathbb{P}}(2)$

I'm reading some notes and I come across this computation in the context of computing Springer fibers: Let $V$ be a $4$ dimensional vector space over $k$ with standard basis $e_i$. We have a nilpotent ...
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### Does the restriction of $\mathcal{O}(1)$ to a quadric have a square root?

Let $Q^n$ be a smooth $n$-dimensional quadric in $P^{n+1}(\mathbb{C})$. Does the restriction of $\mathcal{O}(1)$ to $Q^n$ have a square root, for any $n \geq 1$? If $n = 1$, then $Q^1$ is ...
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### When a line bundle on a projective variety separates tangent vetor?

In the Huybrecht's book Fourier Mukai transform in Algebraic geometry, in order to prove Bondal and Orlov's results (proposition 4.11), it seems that he uses following things. Suppose $k$ is an ...
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### Given an element $\omega \in H^{(1,1)}(X,\mathbb{C})$ how to construct a line bundle with chern class $\omega$?

Let $X$ be a Kahler complex variety of dimension $1$. Let $\omega\in H^{(1,1)}_{dR}(X,\mathbb{C})$ such that $\int_{X}\omega = 0$. Can we find a line bundle $L$ over $X$ such that $c_1(L) = \omega$? ...
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### How should I understand Kodaira dimension?

Say we have projective variety $X$. Its Kodaira dimension $\kappa(X)$ is defined by the “growth exponential” of $P_d := \dim H^0(X,K_X^{\otimes d})$ with respect to $d$, i.e. $\kappa(X) := -\infty$ ...
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### Relationship between the ideal of an effective Cartier divisor and its invertible sheaf

Let $X$ be a scheme. We'll assume it's noetherian to avoid any pathologies. Let $D$ be an effective Cartier divisor on $X$. I am having trouble understanding how to go between the language of ...
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### Explicit description of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a line bundle

I understand the construction of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a sheaf on $\Bbb{P}_\Bbb{C}^1$, but I'm trying to understand how exactly does this define a line bundle and why people call this the &...
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### What is the connection between divisors and line bundles? [duplicate]

I'm somewhat new to algebraic geometry. I'm currently studying algebraic curves primarily over closed fields, (for future discussion let's just call such a curve C). I was taught to think of things ...
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### Relation between tautological line bundle and blow up at the origin

We can define the projective $n$-space $\mathbb{P}^n$ as the quotient of $\mathbb{C}^{n+1}\setminus \{0\}$ by the action of $\mathbb{C}^*$ with all weights equal to $1$. Moreover we can define the ...
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### Local class group and Class group of localizations

Let $R$ be a Noetherian normal domain with divisor class group $Cl(R)$ and Picard group $Pic(R)$ and we can consider $Pic(R)$ as a subgroup of $Cl(R)$. Consider the following two statements: (1) There ...
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### When does the Picard group embeds inside the divisor class group?

Let $(X, \mathcal O_X)$ be a Noetherian, separated, integral scheme that is locally regular in codimension $1$ (i.e. if $\dim \mathcal O_{X,x}=1$ then $\mathcal O_{X,x}$ is regular). Then, is it true ...
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### Prove is a line bundle

I want to prove that a specific object is a line bundle. Consider a normal variety $X$ and let $E$ be a line bundle on $X$. Denote by $s:X\to E$ the zero section, and consider F=(E\setminus s(X))\...
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### holomorphic section of positive line bundle

I read the following statement from the book "L^2 approaches in several complex variables" page 206: Positive dimensional analytic sets must intersect with the zeros of holomorphic sections of ...
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### Identify a pullback line bundle on $\mathbb{P}^1$

Consider a degree $d$ map $f:\mathbb{P}^1 \to \mathbb{P}^m$ for $d \geq 1$ and $m \geq 2$, together with a line bundle $\mathcal{O}_{\mathbb{P}^m}(l)$ over $\mathbb{P}^m$ for $l \geq 1$. Then we have ...
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### Direct proof for $\mathcal{O}_{\mathbb{CP}_{1}}(-2) \cong T^{*}\mathbb{CP}_{1}$

Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the ...
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### Polarization of abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
### Canonical bundle and canonical divisor in a $K3$ surface
The algebraic definition of a $K3$ surface is this: A smooth algebraic suface $X$ is called $K3$ if: i) $X$ has trivial canonical bundle; ii) $h^1(X,\mathcal{O}_X)=0$. I know that i) means that the ...